Compensating for frequency change in flowmeters

ABSTRACT

A method for calculating a property of a fluid includes inducing motion in a conduit such that the conduit oscillates in a first mode of vibration and a second mode of vibration. The first mode of vibration has a corresponding first frequency of vibration and the second mode of vibration has a corresponding second frequency of vibration. The conduit contains a fluid. The first frequency of vibration is determined, and the second frequency of vibration is determined. A property of the fluid based on the first frequency of vibration and the second frequency of vibration is calculated.

CROSS-REFERENCE TO RELATED APPLICATIONS

The application claims priority of U.S. Provisional Application Ser. No.60/772,580, titled COMPENSATING FOR CHANGE IN FREQUENCY OF CORIOLISFLOWMETERS WITH MASS FLOW, and filed on Feb. 13, 2006, and to U.S.Provisional Application Ser. No. 60/827,845, titled COMPENSATING FORCHANGE IN FREQUENCY OF CORIOLIS FLOWMETERS WITH MASS FLOW, and filed onOct. 2, 2006, both of which are incorporated by reference.

TECHNICAL FIELD

The description relates to flowmeters.

BACKGROUND

Flowmeters provide information about materials being transferred througha conduit. For example, mass flowmeters provide a measurement of themass of material being transferred through a conduit. Similarly, densityflowmeters, or densitometers, provide a measurement of the density ofmaterial flowing through a conduit. Mass flowmeters also may provide ameasurement of the density of the material.

For example, Coriolis-type mass flowmeters are based on the Corioliseffect, in which material flowing through a rotating conduit is affectedby a Coriolis force and therefore experiences an acceleration. ManyCoriolis-type mass flowmeters induce a Coriolis force by sinusoidallyoscillating a conduit about a pivot axis orthogonal to the length of theconduit. In such mass flowmeters, the Coriolis reaction forceexperienced by the traveling fluid mass is transferred to the conduititself and is manifested as a deflection or offset of the conduit in thedirection of the Coriolis force vector in the plane of rotation.

In one general aspect, motion is induced in a conduit such that theconduit oscillates in a first mode of vibration and a second mode ofvibration. The first mode of vibration has a corresponding firstfrequency of vibration and the second mode of vibration has acorresponding second frequency of vibration. The conduit contains afluid. The first frequency of vibration is determined, and the secondfrequency is determined. A property of the fluid is determined based onthe first frequency of vibration and the second frequency of vibration.

Implementations may include one or more of the following features. Forexample, the property of the fluid may the density of the fluid. Thedensity may be calculated using:${\rho_{0} = {\frac{1}{2}\left( {D_{4} + E_{4} + \sqrt{\frac{4D_{2}E_{2}}{\omega_{1}^{2}\omega_{2}^{2}} + \left( {D_{4} - E_{4}} \right)^{2}}} \right)}},$where ω₁ is the first frequency, ω₂ is the second frequency, and D₂, D₄,E₂, and E₄ are calibration constants related to physical properties ofthe conduit. A first temperature and a second temperature may bedetermined. The calibration constants may be determined based on thefirst and second temperatures such that the calculation of the densityis compensated for the affect of a differential between the firsttemperature and the second temperature.

The property of the fluid may be the mass flow rate of the fluid.Calculating the mass flow rate may include measuring the oscillation ofthe conduit at a first location along the conduit and at a secondlocation along the conduit. A phase difference based on the oscillationat the first location and the oscillation at the second location may bedetermined. The mass flow rate may be calculated using:${{\overset{.}{m}}_{corrected} = {{{Mfact} \cdot {K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}_{nom} \cdot \frac{1}{\omega_{2}}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {k_{m}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}},$where ω₁ is the first frequency, ω₂ is the second frequency, ω₁₀ is thefirst frequency during a zero-flow condition, ω₂₀ is the secondfrequency during the zero-flow condition, k and k_(m) are calibrationconstants related to physical properties of the conduit, and φ is thephase difference. Mfact may be determined using:${Mfact} = {\frac{\left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)}{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)_{nom}}.}$

Determining the first frequency and the second frequency may includereceiving at least one signal from a sensor configured to monitor theoscillation of the conduit. Determination of the first frequency and thesecond frequency may includes determining the first frequency and thesecond frequency based on the at least one sensor signal.

Determining the first or second frequency may include determiningphysical properties associated with the conduit, and determining anexpression associated with one of the first frequency or the secondfrequency based on the physical properties. The expression may be usedto determine one of the first frequency or the second frequency.

The first frequency may be determined based on the second frequency bycalculating:${\omega_{1}^{2} = {\omega_{2}^{2}\frac{\left( {E_{2} + {\left( {E_{2} - D_{2}} \right)t_{2}}} \right)^{2}}{\left( {E_{2} + {\left( {D_{4} - E_{4}} \right)t_{2}\omega_{2}^{2}}} \right)\left( {D_{2} + {\left( {D_{4} - E_{4}} \right)\left( {1 + t_{2}} \right)\omega_{2}^{2}}} \right)}}},$where ω₁ is the first frequency, ω₂ is the second frequency,$t_{2} = {k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}$where φ is the phase difference k, D₂, D₄, E₂, and E₄ are calibrationconstants related to physical properties of the conduit.

The first mode may be a Coriolis mode, and the second mode may be adriven mode.

In another general aspect, motion is induced motion in a conduit suchthat the conduit oscillates in a first mode of vibration and a secondmode of vibration. The first mode of vibration has a corresponding firstfrequency of vibration and the second mode of vibration has acorresponding second frequency of vibration. The first and the secondfrequencies vary with temperature. The conduit contains a fluid. A firsttemperature that influences the first mode of vibration is determined. Asecond temperature that influences the second mode of vibration isdetermined. A property of the fluid is determined based on the firsttemperature and the second temperature. The property depends on at leastone of the first frequency or the second frequency, and the property isdetermined based on the first and second temperatures such that theproperty is compensated for at least one of the variation of the firstfrequency with temperature or the variation of the second frequency withtemperature.

Implementations may include one or more of the following features. Forexample, the property may depend on both the first and secondfrequencies, and the property may be determined such that it iscompensated for both the variation of the first frequency withtemperature and the variation of the second frequency with temperature.

The property of the fluid may be the density of the fluid. Densitycalibration constants may be determined based on physical properties ofthe conduit. A reference temperature may be determined. The densitycalibration constants may be compensated based on the first temperature,the second temperature, and the reference temperature. The density maybe determined based on the compensated density calibration constants. Inanother example, the density may be determined based on the calibrationconstants and the first frequency or the second frequency. In yetanother example, the density may determined based on the calibrationconstants, the first frequency, and the second frequency.

The property of the fluid may be the mass flow rate of the fluid. Themass flow rate of the fluid may depend on the first frequency during azero-flow condition and the second frequency at the zero-flow condition.

The first temperature may be a temperature of the fluid contained in theconduit, and the second temperature may be a temperature of the conduit.In another example, the first temperature may be a temperature of thefluid contained in the conduit, and the second temperature may be anambient temperature in the vicinity of the conduit. In yet anotherexample, the first temperature may be a temperature of the fluidcontained in the conduit, and the second temperature may be atemperature of a housing of the conduit. In yet another example, thefirst temperature may be a temperature of one location on the conduit,and the second temperature may a temperature of another location on theconduit.

The first mode is a Coriolis mode, and the second mode is a driven mode.

In another general aspect, motion is induced in a conduit such that theconduit oscillates in a first mode of vibration and a second mode ofvibration. The first mode of vibration has a corresponding firstfrequency of vibration and the second mode of vibration has acorresponding second frequency of vibration. The conduit contains afluid. At least one of the first frequency of vibration or the secondfrequency of vibration is determined. A phase difference between themotion of the conduit at a first point of the conduit and the motion ofthe conduit at a second point of the conduit is determined. A quantitybased on the phase difference and the determined frequency isdetermined. The quantity includes a ratio between the first frequencyduring a zero-flow condition and the second frequency during thezero-flow condition. A property of the fluid is determined based on thequantity.

Implementations may include one or more of the following features. Forexample, the quantity may be determined by:${{\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} = \frac{{\left( {D_{4} - E_{4}} \right)\omega_{2}^{2}} + D_{2} - E_{2}}{{\left( {D_{4} - E_{4}} \right)\omega_{2}^{2}k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}} + E_{2}}},$where ω₂ is the second frequency, ω₁₀ is the first frequency during thezero-flow condition, ω₂₀ is the second frequency during the zero-flowcondition, k, D₂, D₄, E₂, and E₄ are calibration constants related tophysical properties of the conduit, and φ is the phase difference.

The property may include a density of the fluid contained in theconduit. Density calibration constants may be determined based onphysical properties of the conduit. A reference temperature may bedetermined. A first temperature that includes the first mode ofvibration and a second temperature that influences the second mode ofvibration may be determined. The density calibration constants may becompensated based on the first temperature, the second temperature, andthe reference temperature. Determining the property may includedetermining the density based on the compensated density calibrationconstants and the quantity. The density may be determined based on:${{\hat{\rho}}_{e} = {\frac{D_{2}}{\omega_{2}^{2}\left( {1 - {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} + D_{4}}},$where ω₂ is the second frequency, ω₁₀ is the first frequency during azero-flow condition, ω₂₀ is the second frequency during the zero-flowcondition, k, D₂ and D₄ are calibration constants related to physicalproperties of the conduit, and φ is the phase difference.

The property may include a mass flow rate of the fluid contained in theconduit. The mass flow rate may be determined based on:${{\overset{.}{m}}_{corrected} = {{{Mfact} \cdot {K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}_{nom} \cdot \frac{1}{\omega_{2}}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {k_{m}\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}},$where K is a factor related to the stiffness of the conduit, ω₂ is thesecond frequency, ω₁₀ is the first frequency during a zero-flowcondition, ω₂₀ is the second frequency during the zero-flow condition,k_(m) is a constant specific to the conduit, φ is the phase difference,and Mfact is${Mfact} = {\frac{\left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)}{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)_{nom}}.}$

The first mode may be a Coriolis mode, and the second mode may be adriven mode.

Implementations of any of the techniques described above may include amethod or process, a system, a flowmeter, or instructions stored on astorage device of flowmeter transmitter. The details of particularimplementations are set forth in the accompanying drawings anddescription below. Other features will be apparent from the followingdescription, including the drawings, and the claims.

DESCRIPTION OF DRAWINGS

FIG. 1A is an illustration of a Coriolis flowmeter using a bentflowtube.

FIG. 1B is an illustration of a Coriolis flowmeter using a straightflowtube.

FIG. 2 is a block diagram of a digital mass flowmeter for measuring aproperty of a fluid.

FIG. 3 is a flowchart illustrating a process for determining a propertyof a fluid contained in a flowtube.

FIG. 4 is a flowchart illustrating a process for determining a propertyof a fluid contained in a flowtube based on two temperaturemeasurements.

FIG. 5 is a flowchart illustrating a process for determining the densityof a fluid based on two temperatures.

FIG. 6 is a flowchart illustrating a process for determining the massflowrate of a fluid based on two temperatures.

FIG. 7 is a flowchart illustrating a process for determining a propertyof a fluid contained in a flowtube.

FIG. 8 is an illustration of a still-walled flowtube.

FIG. 9 is a graph of the poles of a flowtube system as a function ofincreasing mass flow rate.

FIG. 10 is a flowchart illustrating a process for calibrating andconfiguring a flowmeter.

DETAILED DESCRIPTION

Types of flowmeters include digital flowmeters. For example, U.S. Pat.No. 6,311,136, which is hereby incorporated by reference, discloses theuse of a digital flowmeter and related technology including signalprocessing and measurement techniques. Such digital flowmeters may bevery precise in their measurements, with little or negligible noise, andmay be capable of enabling a wide range of positive and negative gainsat the driver circuitry for driving the conduit. Such digital flowmetersare thus advantageous in a variety of settings. For example,commonly-assigned U.S. Pat. No. 6,505,519, which is incorporated byreference, discloses the use of a wide gain range, and/or the use ofnegative gain, to prevent stalling and to more accurately exercisecontrol of the flowtube, even during difficult conditions such astwo-phase flow (e.g., a flow containing a mixture of liquid and gas).

Although digital flowmeters are specifically discussed below withrespect to, for example, FIGS. 1A, 1B and 2, it should be understoodthat analog flowmeters also exist. Although such analog flowmeters maybe prone to typical shortcomings of analog circuitry, e.g., lowprecision and high noise measurements relative to digital flowmeters,they also may be compatible with the various techniques andimplementations discussed herein. Thus, in the following discussion, theterm “flowmeter” or “meter” is used to refer to any type of deviceand/or system in which a Coriolis flowmeter system uses various controlsystems and related elements to measure a mass flow, density, and/orother parameters of material(s) moving through a flowtube or otherconduit.

FIG. 1A is an illustration of a digital flowmeter using a bent flowtube102. Specifically, the bent flowtube 102 may be used to measure one ormore physical characteristics of, for example, a (travelling ornon-travelling) fluid, as referred to above. In FIG. 1A, a digitaltransmitter 104 exchanges sensor and drive signals with the bentflowtube 102, so as to both sense an oscillation of the bent flowtube102, and to drive the oscillation of the bent flowtube 102 accordingly.By quickly and accurately determining the sensor and drive signals, thedigital transmitter 104, as referred to above, provides for fast andaccurate operation of the bent flowtube 102. Examples of the digitaltransmitter 104 being used with a bent flowtube are provided in, forexample, commonly-assigned U.S. Pat. No. 6,311,136.

FIG. 1B is an illustration of a digital flowmeter using a straightflowtube 106. More specifically, in FIG. 1B, the straight flowtube 106interacts with the digital transmitter 104. Such a straight flowtubeoperates similarly to the bent flowtube 102 on a conceptual level, andhas various advantages/disadvantages relative to the bent flowtube 102.For example, the straight flowtube 106 may be easier to (completely)fill and empty than the bent flowtube 102, simply due to the geometry ofits construction. In operation, the bent flowtube 102 may operate at afrequency of, for example, 50-110 Hz, while the straight flowtube 106may operate at a frequency of, for example, 300-1,000 Hz. The bentflowtube 102 represents flowtubes having a variety of diameters, and maybe operated in multiple orientations, such as, for example, in avertical or horizontal orientation.

Referring to FIG. 2, a digital mass flowmeter 200 includes the digitaltransmitter 104, one or more motion sensors 205, one or more drivers210, a flowtube 215 (which also may be referred to as a conduit, andwhich may represent either the bent flowtube 102, the straight flowtube106, or some other type of flowtube), and a temperature sensor 220. Thedigital transmitter 104 may be implemented using one or more of, forexample, a processor, a Digital Signal Processor (DSP), afield-programmable gate array (FPGA), an ASIC, other programmable logicor gate arrays, or programmable logic with a processor core. It shouldbe understood that, as described in U.S. Pat. No. 6,311,136, associateddigital-to-analog converters may be included for operation of thedrivers 210, while analog-to-digital converters may be used to convertsensor signals from the sensors 205 for use by the digital transmitter104.

The digital transmitter 104 generates a measurement of, for example,density and/or mass flow rate of a material flowing through the flowtube215, based at least on signals received from the motion sensors 205. Thedigital transmitter 104 also controls the drivers 210 to induce motionin the flowtube 215. This motion is sensed by the motion sensors 205.

Density measurements of the material flowing through the flowtube arerelated to, for example, the frequency of the motion of the flowtube 215that is induced in the flowtube 215 (typically the resonant frequency)by a driving force supplied by the drivers 210, and/or to thetemperature of the flowtube 215. Similarly, mass flow through theflowtube 215 is related to the phase and frequency of the motion of theflowtube 215, as well as to the temperature of the flowtube 215.

The temperature in the flowtube 215, which is measured using thetemperature sensor 220, affects certain properties of the flowtube, suchas its stiffness and dimensions. The digital transmitter 104 maycompensate for these temperature effects. Also in FIG. 2, a pressuresensor 225 is in communication with the transmitter 104, and isconnected to the flowtube 215 so as to be operable to sense a pressureof a material flowing through the flowtube 215.

It should be understood that both the pressure of the fluid entering theflowtube 215 and the pressure drop across relevant points on theflowtube may be indicators of certain flow conditions. Also, whileexternal temperature sensors may be used to measure the fluidtemperature, such sensors may be used in addition to an internalflowmeter sensor designed to measure a representative temperature forflowtube calibrations. Also, some flowtubes use multiple temperaturesensors for the purpose of correcting measurements for an effect ofdifferential temperature between the process fluid and the environment(e.g., a case temperature of a housing of the flowtube or a temperatureof the flowtube itself). For example, two temperature sensors may beused, one for the fluid temperature and one for the flowtubetemperature, and the difference between the two may be used tocompensate density and/or mass flow calculations as described below.

In FIG. 2, it should be understood that the various components of thedigital transmitter 104 are in communication with one another, althoughcommunication links are not explicitly illustrated, for the sake ofclarity. Further, it should be understood that conventional componentsof the digital transmitter 104 are not illustrated in FIG. 2, but areassumed to exist within, or be accessible to, the digital transmitter104. For example, the digital transmitter 104 will typically includedrive circuitry for driving the driver 210, and measurement circuitry tomeasure the oscillation frequency of the flowtube 215 based on sensorsignals from sensors 205 and to measure the phase between the sensorsignals from sensors 205.

The digital transmitter 104 includes a (bulk) density measurement system240 and a mass flow rate measurement system 250. The bulk densitymeasurement system calculates the density of the travelling fluid, forexample, based on equations (27), (28), (34), (35), or (36) describedbelow, or some variation of either of these equations. The mass flowrate measurement system 250 measures the mass flow rate of thetravelling fluid using, for example, equation (20) described below, orsome variation of this equation. In general, the resonant frequency ofvibration of the flowtube 215 for given fluid changes as the mass flowrate of the fluid changes. This can result in errors in the measureddensity and massflow rate of the fluid if the flowmeter is designedunder the assumption that the resonant frequency only changes with achange in density of the fluid. Using equations (27), (28), (36), or(20), or variations thereof, for the measurement of these items cancompensation for such errors. Furthermore, the ratio of the Coriolismode and driven mode of vibration at zero flow is represented in theseequations. Generally, for some flowtubes, this ratio is fixed, and insuch a situation the fixed value can be used directly. However, in otherflowtubes, this ratio is not fixed. In such situations, equation (32),or a variation thereof, may be used by itself (or combined withequations (27), (28), (36), or (20)), to account for the change in thisratio during operation by using the observed frequency and/or observedphase. In addition, temperature differential compensation describedbelow may be used in combination with any of these techniques.

Referring to FIG. 3, a process 300, which may be implemented by thedigital transmitter 104, may be used to calculate a property of a fluidcontained in the flowtube 215. The process 300 includes inducing motionin the flowtube 215 (302). As described above, the drivers 210 mayinduce motion in the flowtube 215. As a result of the induced motion,the flowtube 215 oscillates in a first mode of vibration and a secondmode of vibration. The first and second modes of vibration havecorresponding frequencies of vibration.

Generally, a ‘bent tube’ Coriolis flowtube that has two drivers can beoperated in either of the first two natural modes of vibration. Theflowtube is forced to oscillate in one ‘driven mode’, and the effect ofCoriolis forces cause a movement in the second ‘Coriolis mode’. Thus,the first mode of vibration may correspond to the ‘Coriolis mode,’ andthe second mode of vibration may correspond to the ‘driven mode.’Alternatively, the first mode of vibration may correspond to the ‘drivenmode,’ and the second mode of vibration may correspond to the ‘Coriolismode.’ The frequency of vibration in the Coriolis mode may be referredto as the Coriolis mode frequency, and the frequency of vibration in thedrive mode may be referred to as the driven mode frequency. In manyimplementations, the driven mode frequency is higher than the Coriolismode frequency. However, physical characteristics of the flowtube 215may cause the Coriolis mode frequency to be higher than the driven modefrequency. In this implementation, the Coriolis mode may correspond tothe second mode of vibration, and the driven mode may correspond to thefirst mode of vibration.

The process 300 also includes determining the first frequency ofvibration (304) and the second frequency of vibration (306). The drivenmode frequency may be observed using the sensor signals from, forexample, sensors 205. The Coriolis mode frequency may be determined in anumber of ways. For instance, with some flowtubes, the Coriolis modefrequency may be directly observed by switching the sense of driverscoupled to the flowtube. This may cause the flowtube to vibrate in theCoriolis mode of operation, which may allow the Coriolis mode frequencyto be observed using the sensor signals from the sensors attached to theflowtube. Also, continuous estimation of the Coriolis frequency may beperformed by analysis of the sensor signals.

Another manner of determining the Coriolis mode frequency may includethe experimental characterization of the flowtube. This may be done toproduce a generalized expression of the Coriolis mode frequency as afunction of flowtube properties such as dimensions, materials, tubethicknesses, fluid and flowtube temperatures, drive frequencies andobserved phase angle/massflow. This expression could use variousmultidimensional curve fitting techniques, such as look-up table,polynomial interpolation or artificial neural nets.

As another alternative, using the analysis shown further below, theCoriolis mode frequency may be calculated from the observed driven modefrequency using the following equation (described further below), whereω₁ is the Coriolis mode frequency and ω₂ is the drive mode frequency:$\omega_{1}^{2} = {\omega_{2}^{2}{\frac{\left( {E_{2} + {\left( {E_{\quad 2} - D_{\quad 2}} \right)t_{\quad 2}}} \right)^{2}}{\left( {E_{2} + {\left( {D_{4} - E_{4}} \right)t_{2}\omega_{2}^{2}}} \right)\left( {D_{2} + {\left( {D_{4} - E_{4}} \right)\left( {1 + t_{2}} \right)\omega_{2}^{2}}} \right)}.}}$

Process 300 also includes calculating a property of the fluid containedin the flowtube 215 based on the first and second frequencies (308). Forexample, the calculated property may be the mass flow rate of the fluid.The following two equations (described further below) may be used todetermine the mass flow rate based on the two frequencies:${Mfact} = \frac{\left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)}{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)_{nom}}$${\overset{.}{m}}_{corrected} = {{{Mfact} \cdot {K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}_{nom} \cdot \frac{1}{\omega_{2}}}\tan\quad{\left( \frac{\varphi}{2} \right) \cdot \left( {1 + {k_{m}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}$

In another example, the property of the fluid may be the density of thefluid. The density may be calculated from the determined frequencies ofvibration using, for example, the following equation (described furtherbelow):${\rho_{0} = {\frac{1}{2}\left( {D_{4} + E_{4} + \sqrt{\frac{4D_{2}E_{2}}{\omega_{1}^{2}\omega_{2}^{2}} + \left( {D_{4} - E_{4}} \right)^{2}}} \right)}},$

where D₂, D₄, E₂, and E₄ are calibration constants, ω₁ is the Coriolismode frequency, and ω₂ is the driven mode frequency. Additionally, theabove density equation is independent of mass flow rate, thus it givesan accurate density calculation regardless of variations in mass flowrate.

Referring to FIG. 4, a process 400, which may be implemented by thedigital transmitter 104, may be used to determine a property of thefluid contained in the flowtube 215 based on two separate temperaturemeasurements that represent a temperature differential. The presence ofa temperature differential may affect the calculations of properties ofthe fluid, such as density and mass flow rate, that depend on at leastone of the frequencies of vibration. Thus, using the two separatetemperature measurements enables the property to be determined based onthe two separate temperature measurements. As a result, the property maybe compensated for at least one of the variation of the first frequencywith temperature or the variation of the second frequency withtemperature.

The temperature differential may be a difference between the temperatureof one part of the flow tube as compared to another. For example, thetemperature of a torsion bar in contact with the flowtube may bedifferent than portions of the flow tube that are in contact with thefluid contained in the flow tube. One of the temperatures may influencethe frequency associated with a particular mode of vibration more thanthe other mode of vibration. For example, the Coriolis mode frequencymay be more influenced by the temperature of the fluid in the flowtube215, and the driven mode frequency may be more influenced by thetemperature of the flow tube body. The temperature that affects thefrequency of a particular mode of vibration may be referred to as the“mode significant temperature.” The effects of the temperaturedifferential on the frequency may be compensated for if the modesignificant temperatures are know. Although it may not be possible tomeasure the mode significant temperatures, as discussed below, othermeasurable temperatures may be used to approximate the mode significanttemperatures.

Similar to process 300, process 400 includes inducing motion in theflowtube 215 (402). The motion may be induced by drivers 210, and themotion may cause the flowtube 215 to oscillate in a first mode ofvibration and a second mode of vibration. As discussed above, the firstmode of vibration may correspond to the Coriolis mode and the secondmode of vibration may correspond to the driven mode. However, asdiscussed above, the first mode of vibration may correspond to thedriven mode, and the second mode of vibration may correspond to theCoriolis mode. The modes of vibration each have correspondingfrequencies of vibration, which may vary with temperature. Unlessaccounted for, this temperature variation may result in inaccuracies inthe calculation of properties that depend on one or more of thefrequencies of vibration, such as density and mass flow rate.

Process 400 also includes determining a first temperature (404), and asecond temperature (406). In general, as described above, there may bemode significant temperatures associated with a flow tube. For instance,a temperature may influence the Coriolis mode of vibration, and atemperature, T₂, may influence the driven mode of vibration. Asmentioned above, measurement of the mode significant temperatures maynot be possible, but they may be approximated by other temperatures. Forexample, the mode significant temperatures may be approximated by alinear combination of the measured temperature of the fluid, T_(f),contained in the flowtube 215 and the ambient temperature or flowtubetemperature, T_(m). The following equation shows a relationship betweenthe mode significant temperatures, T₁ and T₂, the temperature of thefluid contained in the flowtube 215, T_(f), and the flowtubetemperature, T_(m): $\begin{matrix}{{T_{2} - T_{1}} = {{bT}_{f} + {\frac{\left( {1 - b} \right)}{\left( {1 - c} \right)}\left( {T_{m} - {cT}_{f}} \right)} - {aT}_{f} - {\frac{\left( {1 - a} \right)}{\left( {1 - c} \right)}\left( {T_{m} - {cT}_{f}} \right)}}} \\{= {\frac{\left( {b - a} \right)}{\left( {1 - c} \right)}\left( {T_{f} - T_{m}} \right)}}\end{matrix}$

Thus, in one example, the first temperature may be the measuredtemperature of the fluid contained in the flowtube 215, and the secondtemperature may be the temperature of the flowtube 215. In anotherexample, the second temperature may be a temperature representative ofthe environment of the flowtube 215, such as the case temperature of ahousing of the flowtube 215 or the ambient temperature surrounding theflowtube 215. In yet another example, the first temperature may be atemperature of the flowtube 215 at one location along the flowtube 215,and the second temperature may be a temperature of the flowtube 215 atanother location along the flowtube 215. The location along the flowtube215 may be a location on a component that is in contact with, or in thevicinity of, the flowtube 215, such as a torsion bar.

The process 400 also includes determining a property of the fluidcontained in the flowtube 215 based on the first and second temperatures(408). The property of the fluid is one that depends on at least onefrequency of vibration, such as the density of the fluid or its massflow rate, and the property is determined based on the two temperaturessuch that the property is compensated for the variation of the frequencyor frequencies with temperature. For example, the property may becompensated for at least one of the variation of the first frequencywith temperature or the second frequency with temperature. In oneexample, the density of the fluid may depend on the observed driven modefrequency. As a result of this dependency, calculation of the densitymay be affected by changes in the driven mode frequency that result fromtemperature changes. As indicated by the following equation, the densityof the fluid may be calculated using the driven mode frequency and thedensity calibration constants D₂ and D₄, which are functions of thestiffness, dimensions, and enclosed volume of the flowtube 215:$\rho = {\frac{D_{2}}{\omega_{2}^{2}} + {D_{4}.}}$

As described below with respect to FIG. 5, the density calibrationconstants, D₂ and D₄, may be corrected to account for the presence of atemperature differential, which enables the density to be compensated toaccount for the presence of a temperature differential.

In another example, mass flow rate of the fluid may depend on the drivenmode frequency at zero flow and the Coriolis mode frequency at zeroflow, both of which a temperature differential may affect. The mass flowmay be compensated for the presence of the temperature differential, andmay be determined using the following equations (described furtherbelow):${\overset{.}{m}}_{raw} = {{K_{\phi\quad T_{0}}\left( \frac{r}{h^{2}l} \right)}_{r_{0}}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{1}{\omega_{2}}{\tan\left( \frac{\varphi}{2} \right)}}$${\overset{.}{m}}_{tcomp} = {{{\overset{.}{m}}_{raw}\left( {1 - {\beta_{1}\left( {T_{1} - T_{0}} \right)}} \right)}\left( {1 - {k_{i,1}\left( {T_{f} - T_{m}} \right)}} \right)}$

A process for compensating the mass flow rate for the presence of atemperature differential is described in more detail below with respectto FIG. 6.

Referring to FIG. 5, a process 500, which may be implemented by digitaltransmitter 104, may be used for determining the density based on thetwo temperatures determined, for instance, in process 400. Inparticular, process 500 may be used to determine the density based onthe temperature of the flowtube 215 and the temperature of the fluidcontained in the flowtube 215 so as to compensate for the temperaturedifferential between the two. The process 500 includes generating anaugmented flowtube body temperature, T_(m) ^(*) (502). As indicated bythe following equation, the augmented temperature T_(m) ^(*) may becalculated from the measured temperature of the flowtube 215, T_(m), thetemperature of the fluid, T_(f), and an empirically determined constantthat is specific to the flowtube 215, k_(td2):T _(m) ^(*) =T _(m) +k _(td2)(T _(f) −T _(m)).

The process 500 also includes generating temperature-compensated densitycalibration constants, D₂ and D₄, based on the augmented temperature(504). The calibration constants D₂ and D₄ may be temperaturecompensated using the following equations, where T₀ is a referencetemperature and C and D are flowtube specific constants:D ₂ =D ₂₀(1+C(T _(m) ^(*) −T ₀))D ₄ =D ₄₀(1+D(T _(m) ^(*) −T ₀))

Using the temperature-compensated density calibration constants, thedensity of the fluid may be computed (506). For instance, the densitymay be computer based on the density equation described with respect toFIG. 4: $\rho = {\frac{D_{2}}{\omega_{2}^{2}} + {D_{4}.}}$

The temperature-compensated density calibration constants, D₂ and D₄,may enable compensation of the density calculation for the presence ofthe temperature differential.

Although the above density equation depends on the driven modefrequency, ω₂, other implementations are possible. For example, theCoriolis mode frequency may be used to determine the fluid densityinstead of the driven mode frequency. In this case, the density may becalculated based on the following equation, where the calibrationconstants E₂ and E₄ are temperature compensated, similarly to constantsD₂ and D₄: $\rho = {\frac{E_{2}}{\omega_{1}^{2}} + E_{4}}$

More specifically, in this implementation the augmented temperature,T_(f) ^(*) may be based on the fluid temperature rather than thetemperature of the flowtube 215, as shown by the following equation:T _(f) ^(*) =T _(f) −k _(td1)(T _(f) =T _(m)).

The density calibration constants associated with the Coriolis modefrequency, E₂ and E₄, are then compensated using the augmented fluidtemperature to account for the presence of a temperature differential.For example, the calibration constants may be compensated based on thefollowing equations, where E and F are flowtube specific constants:E ₂ =E ₂₀(1+E(T _(f) ^(*) −T ₀))E ₄ =E ₄₀(1+F(T _(f) ^(*) −T ₀))

In addition to compensating the density determination for the presenceof a temperature differential, the density determination mayadditionally be compensated for the effects of mass flow rate on thefrequencies using the following equation:$\rho_{0} = {\frac{1}{2}{\left( {D_{4} + E_{4} + \sqrt{\frac{4D_{2}E_{2}}{\omega_{1}^{2}\omega_{2}^{2}} + \left( {D_{4} - E_{4}} \right)^{2}}} \right).}}$

In the above equation, the calibration constants D₂, D₄, E₂, and E₄ arethe temperature-compensated calibration constants discussed above, ω₁ isthe Coriolis mode frequency, and ω₂ is the driven mode frequency.Because the temperature-compensated calibration constants compensate thedensity determination for the presence of a temperature differential,the above equation enables determination of density compensated for theeffect of the temperature differential on the frequencies and the effectof mass flow rate on the frequencies.

Referring to FIG. 6, a process 600, which may be implemented by thedigital transmitter 104, may be used for determining the mass flow ratebased on the two temperatures determined, for instance, in process 400.In particular, process 600 may be used to determine the mass flow ratebased on the temperature of the flowtube 215 and the temperature of thefluid contained in the flowtube 215 so as to compensate for thetemperature differential between the two. The process 600 includesdetermining the raw mass flow rate {dot over (m)}_(raw) (602). The rawmass flow rate may be determined using the following equation, where 107₂₀ is the driven mode frequency at zero flow, ω₁₀ is the Coriolis modefrequency at zero flow, and φ is the phase difference as measuredbetween the two sensor signals from sensors 205:${\overset{.}{m}}_{raw} = {{K_{\phi\quad T_{0}}\left( \frac{r}{h^{2}l} \right)}_{T_{0}}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{1}{\omega_{2}}{{\tan\left( \frac{\varphi}{2} \right)}.}}$

The above equation determines {dot over (m)}_(raw), which is the massflow rate of the fluid, but this mass flow rate is not compensated toaccount for the presence of the temperature differential.

The process 600 also includes compensating the raw mass flow ratedetermined above for the presence of the temperature differential (604)using the two temperatures. The compensated mass flow rate may bedetermined using the following equation, where {dot over (m)}_(raw) isthe value determined in (602). β₁ is a basic temperature compensationconstant, T₁ is a temperature that is representative of the Coriolismode, T₀ is a reference temperature, k_(td) is a constant that isspecific to the flowtube 215, T_(f) is the temperature of the fluid inthe flowtube 215, and T_(m) is the temperature of the flowtube 215:{dot over (m)} _(tcomp) ={dot over (m)} _(raw)(1−β₁(T ₁ −T ₀))(1−k_(td)(T _(f) −T _(m))).

Additionally, the mass flow rate may be compensated for both the effectsof mass flow rate and the temperature differential on the frequencies byusing the following equation (described further below) to compute {dotover (m)}_(raw) in (602).${\overset{.}{m}}_{raw} = {\frac{K_{\phi}r}{h^{2}l}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{1}{\omega_{2}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {\frac{\omega_{20}^{2}}{\omega_{10}^{2}}\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}$

The results may then by used to determine the temperature compensatedmass flow rate, {dot over (m)}_(tcomp). Determining {dot over (m)}_(raw)in this manner and then using it to compute {dot over (m)}_(tcomp)results in a mass flow rate that is compensated for the presence of atemperature differential and the effects of mass flow rate on thefrequencies.

Referring to FIG. 7, a process 700, which may be implemented by digitaltransmitter 104, may be used to determine a property of a fluidcontained in a conduit such as flowtube 215. In particular, process 700may be used to determine a quantity that includes a ratio of the drivenmode frequency and the Coriolis frequency during a zero-flow condition.This quantity may be used to determine a property of the fluid. Asdescribed above, and further below, the calculation of some propertiesof a fluid, such as the mass flow rate and the density, may involve aquantity that includes a ratio of the driven mode frequency and theCoriolis frequency at zero-flow condition. However, for some flowtubes,or other conditions, this ratio may not be a fixed amount. Therefore, itmay be desirable to calculate the quantity based on observed conditions,such as the driven mode frequency and the phase difference, and then usethe calculated quantity to determine the property.

The process 700 begins by inducing motion in the flowtube 215 such thatthe conduit oscillates in a first mode of vibration and a second mode ofvibration (702). The first mode of vibration may have a frequency ofvibration that corresponding to the Coriolis mode frequency. The secondmode of vibration may have a frequency that corresponds to the drivenmode frequency. The process 700 also includes determining at least oneof the first frequency of vibration or the second frequency of vibration(704). The process 700 further includes determining a phase differencebetween the motion of the flowtube 215 at a first point along theflowtube 215 and the motion of the flowtube 215 at a second point alongthe flowtube 215 (706). The motion at the first and second point may bemeasured, for example, using the motion sensors 205 described above withrespect to FIG. 2

The process 700 also includes determining a quantity based on thedetermined frequency (e.g., the driven mode frequency) and the phasedifference (708). The quantity includes a ratio between the firstfrequency during a zero-flow condition and the second frequency during azero-flow condition, such as the quantity shown on the left-hand side ofthe following equation.${\frac{\omega_{20}^{2\quad}}{\omega_{10}^{2}} - 1} = \frac{{\left( {D_{4} - E_{4}} \right)\omega_{2}^{2}} + D_{2} - E_{2}}{{\left( {D_{4} - E_{4}} \right)\omega_{2}^{2}k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}} + E_{\quad 2}}$

As indicated by the above equation, the quantity may depend on theobserved frequency, the phase difference, and calibration constants,which may be specific to the flowtube 215. In particular, ω₂ is thesecond frequency, ω₁₀ is the first frequency during the zero-flowcondition, ω₂₀ is the second frequency during the zero-flow condition,k, D₂, D₄, E₂, and E₄ are calibration constants related to physicalproperties of the conduit, and φ is the phase difference.

The process 700 also includes determining a property of the fluid basedon the quantity (710). For example, the property of the fluid may bemass flow rate, as shown by the following equation:${Mfact} = \frac{\left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)}{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)_{nom}}$${\overset{.}{m}}_{corrected} = {{{Mfact} \cdot {K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}_{nom} \cdot \frac{1}{\omega_{2}}}\tan\quad{\left( \frac{\varphi}{2} \right) \cdot \left( {1 + {k_{m}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}$

In the above equation, ω₁ is the first frequency, ω₂ is the secondfrequency, ω₁₀ is the first frequency during a zero-flow condition, ω₂₀is the second frequency during the zero-flow condition, k and k_(m) arecalibration constants related to physical properties of the conduit, andφ is the phase difference.

In another example, the property of the fluid may be the density of thefluid, as shown by the following equation where ω₁is the firstfrequency, ω₂ is the second frequency, and D₂, D₄, E₂, and E₄ arecalibration constants related to physical properties of the conduit:${\hat{\rho}}_{e} = {\frac{D_{2}}{\omega_{2}^{2}\left( {1 - {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\quad\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} + {D_{4}.}}$

The rest of this disclosure provides an analytical explanation for theobserved change in vibration frequency of a Coriolis flowmeter with achange in mass flow rate and/or with a temperature differential acrossparts of the system. This explanation can be used to implement thetechniques described above, and further below, to correct for errorsresulting from these effects.

As described above, a ‘bent tube’ Coriolis flowtube that has two driverscan be operated in either of the first two natural modes of vibration.The flowtube is forced to oscillate in one ‘driven mode’, and the effectof Coriolis forces cause a movement in the second ‘Coriolis mode’. Alsoas explained above, the Coriolis mode may be the first mode ofvibration, and the driven mode may be the first mode of vibration.However, the converse may also occur. The change in resonant frequencywith mass flow rate can be explained by considering the effect of thesecondary Coriolis forces resulting from this Coriolis mode superimposedon those in the ‘driven mode’. Furthermore the change in frequency withmass flow rate is shown to lower the natural frequency of lowestfrequency mode, and to raise the upper frequency, irrespective of theshape of the two modes of vibration, or which mode is selected to bedriven.

Mass Flow Rate Affect on Density and Mass Flow Rate Calculations

In the following analysis a simple straight sided rectangular frame tubeis considered and it is assumed that deflections due to applied forceslead to rotations of the tubes only. The deflections and rotations areconsidered to be sufficiently small such that only forces perpendicularto the tubes can be considered. It is believed that similar results willbe obtained by extending the analysis to curved tubes with simplebending modes of vibration.

FIG. 8 shows a schematic of a still walled flowtube. Fluid travels inthe direction ABCD with constant speed v. The side tubes each havelength h and the middle section length l/2. For ease of analysis, theflowtube is shown as straight sided, but the equations should berepresentative of any flowtube with curved sides where motions andforces are resolved in the orthogonal axes. The directions andmagnitudes of the Coriolis acceleration experienced by an element ofmass δm=ρAδl of fluid with density ρ travelling along each straightsection of pipe of cross sectional area A are shown on the diagram. Thisacceleration is due to the change in angular momentum of the particle asits lateral velocity changes when moving to/from the centre of rotation.This fluid acceleration is provided by lateral forces from the tubewall, and therefore the fluid exerts equal and opposite forces on thetube. This theoretical flowtube has two independent modes of rotationabout the x and z axes. The inlet and exit tube sections aresufficiently close to enable cancellation of forces with negligiblemoment. The mass flow is constant and no forces or rotation about the yaxis are considered. Centrifugal/centripetal forces in the xz plane arenot considered—in practice they will cause tube shape bending distortionwhich would affect the modes of vibration.

This model flowtube can be ‘driven’ in either of the two fundamentalmodes of vibration with drives F1 and F2. The motion of the flowtubes isobserved by sensors S1 and S2.

Considering rotation about the x and z axes separately and integratingthe forces along the length of the tubes we obtain the followingequations of motionI _(x){umlaut over (θ)}+C _(θ){dot over (θ)}+K _(θ)θ=(F ₁ −F₂)l/4+2×ρAh×2v{dot over (φ)}×l/4I _(z){umlaut over (φ)}+C _(φ){dot over(φ)}+K _(φ)φ=(F ₁ +F ₂)h ₁ −ρAl/2×2v{dot over (θ)}×h  (1)

where I_(x) and I_(z) are moments of inertia about their respectiveaxes. For uniform pipe and uniform density fluid these terms areproportional to the total mass of flowtube and contents and readilycomputed, but in practice sensor and actuators will add point masseswhich may be compensated by additional masses so this analysis does notexpand them here.

For the small deflections considered, C and K are positive constantsdefining the effective damping and spring stiffness opposing the motion.

Taking Laplace transforms and substituting{dot over (m)}=ρA v  (2)

we obtain the following form of the equations of motion $\begin{matrix}{{\begin{pmatrix}{{I_{x}s^{2}} + {C_{\theta}s} + K_{\theta}} & {{- {hl}}\quad\overset{.}{m}s} \\{{hl}\quad\overset{.}{m}s} & {{I_{x}s^{2}} + {C_{\phi}s} + K_{\phi}}\end{pmatrix}\begin{pmatrix}\theta \\\phi\end{pmatrix}} = \begin{pmatrix}{\left( {F_{1} - F_{2}} \right){l/4}} \\{\left( {F_{1} + F_{2}} \right)h_{1}}\end{pmatrix}} & (3)\end{matrix}$

The response of the system may be expressed as two second orderresponses with modes defined by the solution of the characteristicequation formed from the determinant of the left-hand matrix:—(I _(x) s ² +C _(θ) s+K _(θ))(I _(z) s ² +C _(φ) s+K _(φ))+h ² l ² {dotover (m)} ² s ²=0  (4)

which shows how the modes of vibration are independent when the massflow is zero, but are coupled with non-zero mass flow.

Equation (4) can be solved numerically for any actual values of theconstants, or simple approximate solutions are given later—but thebehaviour of the roots can be illustrated with a classical root locusdiagram, which is shown in FIG. 9. Specifically, FIG. 9 shows the rootlocus of the position of the poles of the system with increasingmass-flow. The damped natural frequency is given by the imaginaryco-ordinate; the undamped natural frequency is given by the distant froma pole to the origin.

With no damping, equation (4) reduces to that of steady oscillation withfrequency found using s=jω such that $\begin{matrix}{{{{\left( {K_{\theta} - {I_{x}\omega^{2}}} \right)\left( {K_{\phi} - {I_{z}\omega^{2}}} \right)} - {h^{2}l^{2}{\overset{.}{m}}^{2}\omega^{2}}} = 0}{Or}} & (5) \\{{{{\left( {\omega^{2} - \omega_{20}^{2}} \right)\left( {\omega^{2} - \omega_{10}^{2}} \right)} - \frac{h^{2}l^{2}{\overset{.}{m}}^{2}\omega^{2}\omega_{10}^{2}\omega_{20}^{2}}{K_{\theta}K_{\phi}}} = 0}{where}} & (6) \\{{\omega_{10}^{2} = \frac{K_{\phi}}{I_{z}}},\quad{\omega_{20}^{2} = \frac{K_{\theta}}{I_{x}}}} & (7)\end{matrix}$

If the solutions to equation (6) are ω=ω₁,ω₂ then it is possible toexpress equation (6) in factorized form:—(ω²−ω₁ ²)(ω²−ω₂ ²)=0  (8)

by expanding and comparing coefficients of powers of ω it is clear thatsolutions to equation (5) have the property (without furtherapproximation) that $\begin{matrix}{{\omega_{1}^{2}\omega_{2}^{2}} = {\frac{K_{\theta}K_{\phi}}{I_{x}I_{z}} = {\omega_{10}^{2}\omega_{20}^{2}}}} & (9)\end{matrix}$

independent of massflow, which suggests that the geometric mean of thefrequencies would be a mass flow independent frequency which could bevery useful for density measurement. Also $\begin{matrix}{{\omega_{1}^{2} + \omega_{2}^{2}} = {{\frac{K_{\phi}}{I_{z}} + \frac{K_{\theta}}{I_{x}} + \frac{h^{2}l^{2}{\overset{.}{m}}^{2}}{I_{x}I_{z}}} = {\omega_{10}^{2} + \omega_{20}^{2} + \frac{h^{2}l^{2}{\overset{.}{m}}^{2}}{I_{x}I_{z}}}}} & (10)\end{matrix}$

Exact solutions to equation (6) are given by $\begin{matrix}{\omega^{2} = {\frac{\omega_{10}^{2} + \omega_{20}^{2}}{2} + \frac{\begin{pmatrix}{{h^{2}l^{2}{\overset{.}{m}}^{2}\omega^{2}\omega_{10}^{2}\omega_{20}^{2}} \mp} \\\sqrt{\begin{matrix}{\left( {{K_{\theta}{K_{\phi}\left( {\omega_{10}^{2} + \omega_{20}^{2}} \right)}} + {h^{2}l^{2}{\overset{.}{m}}^{2}\omega^{2}\omega_{10}^{2}\omega_{20}^{2}}} \right)^{2} -} \\{4K_{\theta}^{2}K_{\phi}^{2}\omega_{10}^{2}\omega_{20}^{2}}\end{matrix}}\end{pmatrix}}{2K_{\theta}K_{\phi}}}} & (11)\end{matrix}$

Assuming that the change in frequency due to mass flow is very small,and that the natural frequencies are distinct, equation (6) will haveapproximate solutions, neglecting terms in {dot over (m)}⁴ and above$\begin{matrix}{{\omega_{1}^{2} \approx \frac{\omega_{10}^{2}}{1 + \frac{h^{2}l^{2}\omega_{20}^{2}{\overset{.}{m}}^{2}}{K_{\phi}{K_{\theta}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}}}}{\omega_{2}^{2} \approx \frac{\omega_{20}^{2}}{1 - \frac{h^{2}l^{2}\omega_{20}^{2}{\overset{.}{m}}^{2}}{K_{\phi}{K_{\theta}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}}}}} & (12)\end{matrix}$

which shows that if ω₂₀>ω₁₀ then ω₂>ω₂₀ and ω₁<ω₁₀ and conversely ifω₂₀<ω₁₀ then ω₂<ω₂₀ and ω₁>ω₁₀.

Now consider the flowtube driven in the second mode of vibration ω=ω₂,with F₂=F₁ just sufficient to counter the effects of damping thenequation (3) has the following steady state time solution, ignoringdamping and free vibration in the Coriolis mode— $\begin{matrix}{{\theta = {\theta_{0}{\sin\left( {\omega_{2}t} \right)}}}{\phi = {{{- \frac{{hl}\quad\overset{.}{m}\omega_{2}}{K_{\phi} - {I_{z}\omega_{2}^{2}}}}\theta_{0}{\cos\left( {\omega_{2}t} \right)}} = {\frac{{hl}\quad\overset{.}{m}\omega_{2}}{K_{\phi}\left( {\frac{\omega_{2}^{2}}{\omega_{10}^{2}} - 1} \right)}\theta_{0}\cos\quad\left( {\omega_{2}t} \right)}}}} & (13)\end{matrix}$assuming all deflections are small, and sensors detect the lineardisplacement S₁,S₂ at their location defined by by r,hS ₁ =hφ+rθS ₂ =hφ−rθ  (14)

The phase difference φ between the signals will be given by$\begin{matrix}{{\tan\left( \frac{\varphi}{2} \right)} = {\frac{h^{2}l\quad\omega_{2}}{K_{\phi}{r\left( {\frac{\omega_{2}^{2}}{\omega_{10}^{2}} - 1} \right)}}\overset{.}{m}}} & (15)\end{matrix}$which can be rearranged to give the familiar Coriolis mass flowmeterform $\begin{matrix}{{\overset{.}{m} = {\frac{K_{\phi}r}{h^{2}l}\left( {\frac{\omega_{2}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{1}{\omega_{2}}{\tan\left( \frac{\varphi}{2} \right)}}}{Now}} & (16) \\\begin{matrix}{\left( {\frac{\omega_{2}^{2}}{\omega_{10}^{2}} - 1} \right) = {{\frac{\omega_{20}^{2}}{\omega_{10}^{2}}\frac{1}{1 - \frac{h^{2}l^{2}\omega_{20}^{2}{\overset{.}{m}}^{2}}{K_{\phi}{K_{\theta}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}}}} - 1}} \\{\approx {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1 + {\frac{\omega_{20}^{2}}{\omega_{10}^{2}}\frac{h^{2}l^{2}\omega_{20}^{2}{\overset{.}{m}}^{2}}{K_{\phi}{K_{\theta}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}}}}} \\{\approx {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\left( {1 + {\frac{\omega_{20}^{2}}{\omega_{10}^{2}}\frac{h^{2}l^{2}\omega_{20}^{2}{\overset{.}{m}}^{2}}{K_{\phi}{K_{\theta}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}^{2}}}} \right)}} \\{\approx {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\left( {1 + {\frac{\omega_{20}^{2}}{\omega_{10}^{2}}\frac{K_{\phi}}{K_{\theta}}\frac{\quad r^{\quad 2}}{\quad h^{\quad 2}}\left( \frac{h^{2}l\quad\omega_{20}{\overset{.}{m}}^{2}}{K_{\phi}{r\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}} \right)^{2}}} \right)}}\end{matrix} & (17)\end{matrix}$

Given that the change in frequency is small let ω₂₀≈ω₂ in the right handterm and substitute for the massflow from equation (16) to obtain$\begin{matrix}{{\left( {\frac{\omega_{2}^{2}}{\omega_{10}^{2}} - 1} \right) = {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\left( {1 + {\frac{\omega_{20}^{2}}{\omega_{10}^{2}}\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}{Or}} & (18) \\\begin{matrix}{\left( {\frac{\omega_{2}^{2}}{\omega_{10}^{2}} - 1} \right) = {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\left( {1 + {\frac{I_{z}}{I_{x}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}} & \quad\end{matrix} & (19)\end{matrix}$

So that equation (16) for the mass flow as a function of phase angle φand frequency ω₂, all other variables assumed constant, taking intoaccount the frequency change with mass flow can be expressed as$\begin{matrix}{{\overset{.}{m} = {\frac{K_{\phi\quad r}}{h^{2}l}\left( {\frac{\quad\omega_{20}^{2}}{\quad\omega_{10}^{2}} - 1} \right)\frac{1}{\omega_{2}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {\frac{\omega_{20}^{2}}{\omega_{10}^{2}}\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}}{Similarly}} & (20) \\\begin{matrix}{{\omega_{2}^{2}\left( {1 - \frac{h^{2}l^{2}\omega_{20}^{2}{\overset{.}{m}}^{2}}{K_{\phi}{K_{\theta}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}}} \right)} = {\omega_{2}^{2}\begin{pmatrix}{{1 -}\quad} \\{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}\left( \frac{\omega_{20}h^{2}l\quad\overset{.}{m}}{K_{\phi}\quad{r\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}} \right)^{2}}\end{pmatrix}}} \\{\approx {\omega_{2}^{2}\left( {1 - {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}\end{matrix} & (21)\end{matrix}$

Which by comparison with equation (12) implies that $\begin{matrix}{\omega_{20}^{2} = {\omega_{2}^{2}\left( {1 - {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}} & (22)\end{matrix}$

Now the frequency of oscillation of the flowtube with zero flow ω₂₀ willbe a function of the fluid density ρ and the fixed mass of the flowtubewith an expression such as $\begin{matrix}\begin{matrix}{\omega_{20}^{2} = \frac{K_{\theta}}{I_{x}}} \\{= \frac{K_{\theta}}{M_{f} + {b\quad\rho}}}\end{matrix} & (23)\end{matrix}$

Where M_(t) is a constant term representing the fixed flowtube mass anddimensions, b is another constant derived from flowtube dimensions whichshows how the variable mass of the fluid contained in the flowtube maybe incorporated.

So the flowtube can be calibrated to give $\begin{matrix}{\rho_{0} = {\frac{D_{2}}{\omega_{20}^{2}} + D_{4}}} & (24)\end{matrix}$

Where D₂,D₄ are flowtube specific calibration constants, which may befurther compensated for the effects of temperature.

An estimate of the true fluid density can be obtained with$\begin{matrix}{\rho_{e} = {\frac{D_{2}}{\omega_{2}^{2}} + {D_{4}.}}} & (25)\end{matrix}$

So that the error is given by $\begin{matrix}\begin{matrix}{{\rho_{e} - \rho_{0}} = {{- \frac{D_{2}}{\omega_{20}^{2}}}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \\{= {{- \left( {\rho_{0} - D_{4}} \right)}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}}\end{matrix} & (26)\end{matrix}$

Note that the equation (26) D₄ is negative, so that when the “CoriolisMode” is less than the “Drive Mode” ω₂₀, an uncorrected density readingwill under-read, for other tubes where the frequency ratio is reversedthey will over-read. Equation (20) predicts that irrespective of themagnitude of the ratio of frequencies an uncorrected massflow readingwill under-read a high flow/phase angle. Note also that equations (16)and (20) show that the phase signal for a given massflow (all otherconnections similar) changes sign depending on the location of theCoriolis frequency above or below the Drive frequency.

Substituting equation (22) into equation (24) we obtain a version ofequation (25) which provides a mass flow insensitive estimate of thefluid density {circumflex over (ρ)}_(e) directly from the observedflowtube drive frequency ω₂ and the observed phase angle φ.$\begin{matrix}{{\hat{\rho}}_{e} = {\frac{D_{2}}{\omega_{2}^{2}\left( {1 - {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} + D_{4}}} & (27)\end{matrix}$

Note that we can re-arrange equation (26) to give an improved estimateof the true fluid density {circumflex over (ρ)}₀ using the uncorrecteddensity estimate ρ_(e) from equation (25) $\begin{matrix}{{\overset{̑}{\rho}}_{0} = \frac{\rho_{e} - {{D_{4}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}}{\left( {1 - {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}} & (28)\end{matrix}$

Note also that the zero flow natural frequency of the Coriolis mode ω₁₀will also be a function of the true fluid density ρ₀, and in a similarmanner to equation (24) may be calibrated as $\begin{matrix}{\rho_{0} = {\frac{E_{2}}{\omega_{10}^{2}} + E_{4}}} & (29)\end{matrix}$

Combining equations (29) and (24) the ratio of the frequencies is givenby $\begin{matrix}{\frac{\omega_{20}^{2}}{\omega_{10}^{2}} = \frac{{\left( {D_{4} - E_{4}} \right)\omega_{20}^{2}} + D_{2}}{E_{2}}} & (30)\end{matrix}$

Substituting an estimate for ω₂₀ in the right hand side, based on theobserved ω₂ using equation (22) we obtain $\begin{matrix}{\frac{\omega_{20}^{2}}{\omega_{10}^{2}} = \frac{{\left( {D_{4} - E_{4}} \right){\omega_{2}^{2}\left( {1 - {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}} + D_{2}}{E_{2}}} & (31)\end{matrix}$

Which simplifies to $\begin{matrix}{{\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} = \frac{{\left( {D_{4} - E_{4}} \right)\omega_{2}^{2}} + D_{2} - E_{2}}{{\left( {D_{4} - E_{4}} \right)\omega_{2}^{2}\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}} + E_{2}}} & (32)\end{matrix}$

Which may be further simplified to $\begin{matrix}{{\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} = \frac{{\left( {D_{4} - E_{4}} \right)\omega_{2}^{2}} + D_{2} - E_{2}}{{\left( {D_{4} - E_{4}} \right)\omega_{2}^{2}k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}} + E_{2}}} & \left( {32a} \right)\end{matrix}$

In equation (32a), ω₂ is the second frequency, ω₁₀ is the firstfrequency during the zero-flow condition, ω₂₀ is the second frequencyduring the zero-flow condition, k, D₂, D₄, E₂, and E₄ are calibrationconstants related to physical properties of the conduit, and φ is thephase difference.

Therefore we have an expression for this ratio as a function of observedfrequency and phase, and calibration constants. This can be used inequations for mass flow and density.

The above analysis also may be used to obtain a massflow independentmeasure of the fluid density based on the observed driven frequency andthe natural frequency of the Coriolis mode.

Repeating equation (9) $\begin{matrix}{{\omega_{1}^{2}\omega_{2}^{2}} = {\frac{K_{\theta}K_{\phi}}{I_{x}I_{z}} = {\omega_{10}^{2}\omega_{20}^{2}}}} & (33)\end{matrix}$

Which indicates that the product (or geometric mean) of the Coriolis andDrive frequencies is a function only on fluid density, and not massflow.

Now by experiment at zero flow, with different fluids in the flowtube itis possible to obtain calibration coefficients which explain how eachfrequency varies with fluid density, repeating equations (24) and (29)$\begin{matrix}{\rho_{0} = {\frac{D_{2}}{\omega_{20}^{2}} + D_{4}}} & (34) \\{\rho_{0} = {\frac{\quad E_{\quad 2}}{\quad\omega_{\quad 10}^{\quad 2}} + E_{\quad 4}}} & (35)\end{matrix}$

Combining (34) and (35) we may obtain an expression for the fluiddensity as a function of the actual frequencies for any massflow.$\begin{matrix}{\rho_{0} = {\frac{1}{2}\left( {D_{4} + E_{4} + \sqrt{\frac{4\quad D_{2}E_{2}}{\omega_{1}^{2}\omega_{2}^{2}} + \left( {D_{4} - E_{4}} \right)^{2}}} \right)}} & (36)\end{matrix}$

Now ω₂ is the observed driven frequency of the flowtube, and ω₁ is thenatural frequency of the Coriolis mode at the actual mass flowrate.

As described above, techniques for measuring the Coriolis frequencyinclude, but are not limited to the following:

1. With some flowtubes, direct observation by switching the sense ofdrivers coupled to the flowtube. This may cause the flowtube to vibratein the Coriolis mode of operation.

2. Continuous estimation of the Coriolis frequency may be performed byanalysis of the sensor signals.

3. Experimental characterization of the flowtube may be performed toproduce a generalized expression of the Coriolis frequency as a functionof flowtube properties such as dimensions, materials, tube thicknesses,fluid and flowtube temperatures, drive frequencies and observed phaseangle/massflow. This expression could use various multidimensional curvefitting techniques, such as look-up table, polynomial interpolation orartificial neural nets.

4. Using the analysis shown previously in this document, experimentalcalibration of coefficients may lead to an ability to calculate thenatural frequency of the Coriolis mode based on the observed driven modefrequency. For example:—

Substituting $\begin{matrix}{t_{2} = {\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} & (37)\end{matrix}$

An expression for ω₁ may be obtained as follows $\begin{matrix}{\omega_{1}^{2} = {\omega_{2}^{2}\quad\frac{\left( {E_{2} + {\left( {E_{2} - D_{2}} \right)t_{2}}} \right)^{2}}{\left( {E_{2} + {\left( {D_{4} - E_{4}} \right)t_{2}\omega_{2}^{2}}} \right)\left( {D_{2} + {\left( {D_{4} - E_{4}} \right)\left( {1 + t_{2}} \right)\omega_{2}^{2}}} \right)}}} & (38)\end{matrix}$

Now for a class of flowtubes where the ratio of zero mass flowfrequencies are constant irrespective of the fluid density we have$\begin{matrix}{{D_{4} = E_{4}}{\frac{D_{2}}{E_{2}} = {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} = \gamma}}} & (39)\end{matrix}$

An equation (38) reduces to $\begin{matrix}\begin{matrix}{\omega_{1}^{2} = {\omega_{2}^{2}\quad\frac{\left( {E_{2} + {\left( {E_{2} - D_{2}} \right)t_{2}}} \right)^{2}}{E_{2}D_{2}}}} \\{= {\frac{\omega_{2}^{2}}{\gamma}\left( {1 + {\left( {1 - \gamma} \right)\quad t_{2}}} \right)^{2}}}\end{matrix} & (40)\end{matrix}$

Temperature Compensation with Multiple Temperature Measurements

In addition to the effect of mass flow on the frequency, the temperaturedifferential between the fluid and the flowtube may affect the densityand/or mass flow calculations. Implementations may compensate for thisaffect of the temperature differential.

A simplified version of equation (20), ignoring the effect of mass flowon frequency, relating phase angle to mass flow is— $\begin{matrix}{\overset{.}{m} = {\frac{K_{\phi}r}{h^{2}l}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\quad\frac{1}{\omega_{2}}{\tan\left( \frac{\varphi}{2} \right)}}} & (41)\end{matrix}$

Where the no-flow Coriolis and Drive mode frequencies are defined by$\begin{matrix}{{\omega_{10}^{2} = \frac{K_{\phi}}{I_{z}}},{\omega_{20}^{2} = \frac{K_{\theta}}{I_{x}}}} & (42)\end{matrix}$

It is to be expected that temperature change will reduce the Young'smodulus of the material used in the flowtube, and increase thedimensions through thermal expansion.

A first order model approximation to these effects isK _(φT) ₁ =K _(φT) ₀ (1−β₁(T ₁ −T ₀))ω_(10 T) ₁ ²=ω_(10 T) ₀ ²(1−α₁(T ₁ −T ₀))ω_(20 T) ₂ ²=ω_(20 T) ₀ ²(1−α₂(T ₂ −T ₀))  (43)

Where T₀ is a reference temperature, T₁,T₂ are representativetemperatures for the Coriolis and Drive vibration modes.

For robust massflow calibration, it is useful to understand thevariation with differential temperature of the ‘ideally constant’multiplicative factor $\begin{matrix}\begin{matrix}{\left( {\frac{\omega_{20\quad T_{2}}^{2}}{\omega_{10\quad T_{1}}^{2}} - 1} \right) = \left( {\frac{\omega_{20\quad T_{0}}^{2}\left( {1 - {\alpha_{2}\left( {T_{2} - T_{0}} \right)}} \right)}{\omega_{10\quad T_{0}}^{2}\left( {1 - {\alpha_{1}\left( {T_{1} - T_{0}} \right)}} \right)} - 1} \right)} \\{\approx \left( {{\frac{\omega_{20\quad T_{0}}^{2}}{\omega_{10\quad T_{0}}^{2}}\left( {1 - {\alpha_{2}\left( {T_{2} - T} \right)}_{0} + {\alpha_{1}\left( {T_{1} - T_{0}} \right)}} \right)} - 1} \right)} \\{\approx {\left( {\frac{\omega_{20\quad T_{0}}^{2}}{\omega_{10\quad T_{0}}^{2}} - 1} \right)\begin{pmatrix}{1 - \frac{\frac{\omega_{20\quad T_{0}}^{2}}{\omega_{10\quad T_{0}}^{2}}}{\left( {\frac{\omega_{20\quad T_{0}}^{2}}{\omega_{10\quad T_{0}}^{2}} - 1} \right)}} \\\left( {{\alpha_{2}\left( {T_{2} - T_{0}} \right)} - {\alpha_{1}\left( {T_{1} - T_{0}} \right)}} \right)\end{pmatrix}}}\end{matrix} & (44)\end{matrix}$

A ‘density optimized’ flowtube will be designed to keep the ratio offrequencies constant with different density fluids and temperatures.Therefore when in therma equilibrium at Temperature T $\begin{matrix}{{T_{1} = {T_{2} = T}}{\alpha_{2} = {\alpha_{1} = \alpha}}} & (45) \\\begin{matrix}{\left( {\frac{\omega_{20T_{2}}^{2}}{\omega_{10T_{1}}^{2}} - 1} \right) \approx {\left( {\frac{\omega_{20T_{0}}^{2}}{\omega_{10T_{0}}^{2}} - 1} \right)\left( {1 - {\frac{\frac{\omega_{20T_{0}}^{2}}{\omega_{10T_{0}}^{2}}}{\left( {\frac{\omega_{20T_{0}}^{2}}{\omega_{10T_{0}}^{2}} - 1} \right)}{\alpha\left( {T_{2} - T_{1}} \right)}}} \right)}} & \quad\end{matrix} & (46)\end{matrix}$

Now it may not be possible to directly measure the appropriatetemperatures corresponding to the mode significant temperatures T₁,T₂,but it is reasonable to assume that in steady-state both, and a flowtubemeasured temperature T_(m) are linear combinations of the fluidtemperature T_(f) and the ambient temperature T_(a),

i.e.T ₁ =aT _(f)+(1−a)T _(a)T ₂ =bT _(f)+(1−b)T _(a)T _(m) =cT _(f)+(1−c)T _(a)  (47)

Where a,b,c are in the range 0-1.

Rearranging to eliminate the ambient temperature we obtain$\begin{matrix}\begin{matrix}{{T_{2} - T_{1}} = {{bT}_{f} + {\frac{\left( {1 - b} \right)}{\left( {1 - c} \right)}\left( {T_{m} - {cT}_{f}} \right)} - {aT}_{f} - {\frac{\left( {1 - a} \right)}{\left( {1 - c} \right)}\left( {T_{m} - {cT}_{f}} \right)}}} \\{= {\frac{\left( {b - a} \right)}{\left( {1 - c} \right)}\left( {T_{f} - T_{m}} \right)}}\end{matrix} & (48)\end{matrix}$

Therefore an appropriate form of the massflow equation with temperaturecompensation is $\begin{matrix}{{{\overset{.}{m}}_{raw} = {{K_{\phi\quad T_{0}}\left( \frac{r}{h^{2}l} \right)}_{T_{0}}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)\frac{1}{\omega_{2}}{\tan\left( \frac{\varphi}{2} \right)}}}{{\overset{.}{m}}_{tcomp} = {{{\overset{.}{m}}_{raw}\left( {1 - {\beta_{1}\left( {T_{1} - T_{0}} \right)}} \right)}\left( {1 - {k_{td}\left( {T_{f} - T_{m}} \right)}} \right)}}} & (49)\end{matrix}$

Where k_(td) is another flowtube specific constant $\begin{matrix}{k_{td} = {\frac{\left( \frac{\omega_{20T_{0}}^{2}}{\omega_{10T_{0}}^{2}} \right)}{\left( {\frac{\omega_{20T_{0}}^{2}}{\omega_{10T_{0}}^{2}} - 1} \right)}\frac{\left( {b - a} \right)}{\left( {1 - c} \right)}\alpha}} & (50)\end{matrix}$

An interesting prediction from equation (50) is that this temperaturedifference factor changes sign depending on whether the drive frequencyis above or below the Coriolis frequency.

Note also that equation (49) includes the basic temperature compensationas a function of temperature T₁. It may be appropriate to approximatethis by T_(f) where, by inspection, the Coriolis mode will be moreinfluenced by fluid temperature than flowtube body temperature T_(m). Inaddition, equation (49) does not take into account the affect of massflow on frequency. However, this may be taken into account using, forexample, the second term in equation (20). For instance, m_(raw) inequation (49) may be calculated based on equation (20), and thentemperature compensated.

This analysis also may be used to compensate density calculations basedon the temperature difference between the fluid temperature and the flowtube temperature. A basic equation for density derived from equation(25) is $\begin{matrix}{\rho = {\frac{D_{2}}{\omega_{2}^{2}} + D_{4}}} & (51)\end{matrix}$

The density calibrations constants D₂ and D₄ are functions of theflowtube stiffness, dimensions and enclosed volume and it is reasonableto assume that they may be characterized as having linear variation withtemperatureD ₂ =D ₂₀(1+C(T ₂ −T ₀))D ₄ =D ₄₀(1+D(T ₂ −T ₀))  (52)

Where C and D are flowtube type specific constants obtained by carefulexperiments with different density fluids with different stable‘equilibrium’ temperatures, where

T₂=T₁ and D₂₀ and D₄₀ are base values at temperature T₀.

In the presence of a temperature differential across the flowtube, theideal temperature to use is T₂, which is assumed to be approximated byT_(m) but may not be able to be directly observed.

In that case, equation (47) can be rearranged to show that$\begin{matrix}{T_{2} = {T_{m} + {\frac{\left( {b - c} \right)}{\left( {1 - c} \right)}\left( {T_{f} - T_{m}} \right)}}} & (53)\end{matrix}$

Therefore a correction for the temperature compensation of the densitymeasurement in the presence of temperature difference may use anaugmented temperature T_(m) ^(*) whereT _(m) ^(*) =T _(m) +k _(td2)(T _(f) −T _(m))  (54)

Where k_(td2) is an empirically determined constant for each flowtubetype.

Therefore an algorithm for computing a density estimate in the presenceof a temperature differential across the flowtube is

a) Generate T_(m) ^(*) from equation (54)

b) Generate temperature compensated D₂ and D₄ from equation (52) usingT_(m) ^(*) as an estimate of T₂D ₂ =D ₂₀(1+C(T _(m) ^(*) −T ₀))D ₄ =D ₄₀(1+D(T _(m) ^(*) −T ₀))  (55)

c) Use equation (51) to obtain an estimate of fluid density from thedrive frequency.

Note that an analogous procedure can be applied when using the Coriolisfrequency to determine the fluid density (per, for example, the equation$\left. {\rho_{0} = {\frac{E_{2}}{\omega_{1}^{2}} + E_{4}}} \right),$the equivalent equations to (53), (54), and (55) are $\begin{matrix}{T_{1} = {T_{f} - {\frac{\left( {1 - a} \right)}{\left( {1 - c} \right)}\left( {T_{f} - T_{m}} \right)}}} & (56) \\{T_{f}^{*} = {T_{f} - {k_{{td}\quad 1}\left( {T_{f} - T_{m}} \right)}}} & (57) \\{{E_{2} = {E_{20}\left( {1 + {E\left( {T_{f}^{*} - T_{0}} \right)}} \right)}}{E_{4} = {E_{40}\left( {1 + {F\left( {T_{f}^{*} - T_{0}} \right)}} \right)}}} & (58)\end{matrix}$

In addition, the temperature difference may be compensated in additionto the compensation for the affect of mass flow on frequency by usingthe temperature compensated coefficients D₂, D₄, E₂, E₄ (calculated, forexample, from equations (55) and (58)) in equation (36).

Meter Calibration and Configuration Process for Temperature and MassFlow Rate Compensation

Referring to FIG. 10, a process 1000 calibrates and configures aflowmeter transmitter to compensate for a temperature differential andthe effect of mass flow rate on the frequency. The process 1000 includesperforming an offline calibration using the following equation todetermine the coefficient K_(d) (1002). The coefficient K_(d) describesthe effect of massflow on frequency, for various massflow rates withphase angle φ: $\begin{matrix}{{\rho_{e} - \rho_{0}} = {{- \left( {\rho_{0} - D_{4}} \right)}\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)K_{d}{{\tan^{2}\left( \frac{\varphi}{2} \right)}.}}} & (59)\end{matrix}$

The coefficient K_(d) is computed using the known zero mass flow valuesof the frequencies ω₂₀ and ω₁₀. The ratio of the zero mass flowfrequencies is a production controlled variable that is specific to eachflowtube type. D₄ is the temperature compensated density calibrationvariable and may be determined from no-flow air/water tests. ρ_(e) isthe density that would be observed without massflow correction, ρ₀ isthe true density at each point.

The process 1000 also includes performing an online compensation toaccount for the effects of massflow on frequency (1004). Using thefollowing equation, discussed above as equation (37),:$t_{2} = {\frac{K_{\phi}}{K_{\theta}}\frac{r^{2}}{h^{2}}{\tan^{2}\left( \frac{\varphi}{2} \right)}}$

with calibration constant K_(d) from above to replace the combination ofconstants, t₂ may be determined as shown in the following equation.$\begin{matrix}{t_{2} = {K_{d}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} & (60)\end{matrix}$

Process 1000 continues by determining the Coriolis mode frequency ω₁(1006) from the following equation:$\omega_{1}^{2} = {\omega_{2}^{2}{\frac{\left( {E_{2} + {\left( {E_{2} - D_{3}} \right)t_{2}}} \right)^{2}}{\left( {E_{2} + {\left( {D_{4} - E_{4}} \right)t_{2}\omega_{2}^{2}}} \right)\left( {D_{2} + {\left( {D_{4} - E_{4}} \right)\left( {1 + t_{2}} \right)\omega_{2}^{2}}} \right)}.}}$

The density is then determined using the Coriolis mode frequency and thetemperature-compensated density coefficients (1008). For example,equation (36) is used with the temperature difference compensateddensity coefficients as per equations (54) and (55), (57) and (58) tocalculate the density.

The process 1100 also includes calculating the mass flow rate (1010)using the following equation, which is a basic massflow equation:$\begin{matrix}{{\overset{.}{m}}_{raw} = {{K_{{VO}\quad 2} \cdot \left( {1 + {A\left( {T_{m} - T_{0}} \right)}} \right) \cdot \frac{1}{\quad\omega_{\quad 2}}}{{\tan\left( \frac{\varphi}{\quad 2} \right)}.}}} & (61)\end{matrix}$

This is then compensated for temperature difference (1012) using thefollowing equation, discussed above as equation (49){dot over (m)} _(tcomp) ={dot over (m)} _(raw).(1−k _(td)(T _(f) −T_(m)))  (62))

This quantity is then compensated for the effects of massflow onfrequency (1014) which give a massflow effect on massflow using equation(20) $\begin{matrix}{{\overset{.}{m}}_{output} = {{\overset{.}{m}}_{tcomp} \cdot \left( {1 + {k_{m}{\tan\left( \frac{\varphi}{2} \right)}}} \right)}} & (63)\end{matrix}$

Where the new coefficient k_(m) is obtained from an offline massflowcalibration procedure at the same time as the basic calibration factorK_(FC2). A and k_(td) are assumed to be flowtube type specifictemperature coefficients which are determined by careful temperaturecontrolled trials on an example flowtube characteristic of the size,wall thickness and construction material.

Additionally, linear compensation of the massflow and density readingsas a function of an additional external pressure measurement may beperformed. Pressure stiffens the flowtube but may also change the ratioof frequencies with corresponding effects on massflow and densityreadings as predicted by the equations in this document.

When using flowtubes in which the ratio of frequencies is not constantwith different fluid densities (as discussed with respect to equation32), the above form of the density equation may be used, but additionalcalibration steps may be needed to determine the calibration of theCoriolis mode frequency/density behaviour for each individual tube.Equation (32) may be used with the various equations for compensatingfor the effect of mass flow on frequency to also take into account thevariation of the ratio of frequencies with density. The same ratio offrequencies also appears in the temperature difference combinations buttrials to determine all the effective constants for compensation may notbe feasible. An alternative approach would be to use software to observethe effective Coriolis frequency online—this would be enhanced if themanufacturing of the flowtubes was optimized to produce a single ‘clean’Coriolis frequency to track. This technique would also help tocompensate for dynamic temperature change effects and the basicasymmetric effect of pressure of the vibrating frequencies as describedfurther below.

Compensation with Coriolis Frequency

As indicated above, observation of the effective Coriolis frequencyonline may be used to compensate for dynamic temperature change effectsand the basic asymmetric effect of pressure of the vibratingfrequencies. Ignoring the effects of massflow on frequency (perfectlyvalid at low flow, and when comparing the same flowrates at differenttemperature conditions)$\overset{.}{m} = {{K\left( {\frac{\omega_{2}^{2}}{\omega_{1}^{2}} - 1} \right)}\frac{1}{\omega_{2}}{\tan\left( \frac{\varphi}{2} \right)}}$where ω₂ is the operating drive frequency and ω₁ is the Coriolis modefrequency, φ the measured phase difference. K is a factor related to thestiffness of the flowtube and the linear dimensions that relate theobserved deflection in the Coriolis made to mass flow.

We expect to be able to characterize the changes in K with flowtubetemperature, and we would expect that in thermal equilibrium the ratioof frequencies would remain constant—Both modes due to the change inYoung's modulus and thermal expansion of similar material. But we nowconsider the case where the ratio of frequencies is not constant, butrather is a function of, for example, fluid density, temperaturedifference, or fluid pressure.

Assuming the basic temperature correction provided for the variation ofK is correct, the above equation predicts that the true massflow will berelated to the apparent massflow by${\overset{.}{m}}_{true} = {\frac{\left( {\frac{\omega_{2}^{2}}{\omega_{1}^{2}} - 1} \right)}{\left\lbrack \left( {\frac{\omega_{2}^{2}}{\omega_{1}^{2}} - 1} \right) \right\rbrack_{0}}{\overset{.}{m}}_{apparent}}$where the subscript 0 refers to the values at flowtube calibrationconditions.

The change in frequency of oscillation with flowrate is describedearlier; both the drive and the Coriolis mode frequencies change by thesame factor in opposite directions (the higher goes up, the lower down).

This change in frequencies (with true massflow) affects the massflowcalibration via the same mechanism as anything else which changes thefrequencies, and results in a characteristic negative error at highflows. A simple correction algorithm has been demonstrated to correctfor this effect, and it is important to understand the relationshipbetween the known change in frequencies due to massflow and the unknownvia temperature difference, fluid density pressure etc.

Theory suggests the true relationship between massflow and phase angleis given by the equation$\overset{.}{m} = {{K\left( {\frac{\omega_{2}^{2}}{\omega_{1}^{2}} - 1} \right)}\frac{1}{\omega_{2}}{\tan\left( \frac{\varphi}{2} \right)}}$where ω₂ is the actual drive frequency and ω₁₀ the Coriolis modefrequency at zero flow.

Applying the correction for the effect of massflow this equation becomes$\overset{.}{m} = {{K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}\frac{1}{\omega_{2}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {k_{m}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}$where ω₂₀ is the equivalent zero flow drive frequency, k_(m) a flowtubespecific constant. This is a convenient form as without differentialtemperature or pressure effects this ratio of frequencies can often beassumed to remain constant.

Now the observed frequencies are related to these zero flow frequenciesvia the theoretical expression considering only the massflow effects$\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right) = \left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)$so a proposed algorithm for using the observed vibration frequencies toautomatically account for temperature difference (or fluid density orpressure) is to use the form${Mfact} = \frac{\left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)}{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)_{nom}}$${\overset{.}{m}}_{correcrted} = {{{Mfact} \cdot {K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}_{nom} \cdot \frac{1}{\omega_{2}}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {k_{m}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}$

Theoretically${k = {\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)k_{d}}},\quad{k_{m} = {\left( \frac{\omega_{20}^{2}}{\omega_{10}^{2}} \right)k_{d}}},$where k_(d) is the coefficient used in the massflow independent densityequation. The relationship between k_(d) and k_(m) can be experimentallyverified to within 10%, the difference explainable in terms of atemperature and pressure variation during trials, which means that thisequation could be simplified. However it is preferably to leave it inthis form as the Mfact term can be heavily filtered to track relativelyslow changes in pressure/temperature only. Keeping k_(m) separate alsoallows a degree of tuning to help match theory/experiment.Code Example

The following is an example of code for implementing some of theforegoing techniques in a Coriolis flowmeter transmitter. /* double tmp;double my_square( double x ) {  return (x*x); } /* Estimate the Coriolismode frequency */  double omegals( double c2, double D2, double D4,double E2, double E4) {  double pd, t2;  pd = phase_stats->getMean( );//use a filtered version of phase difference  t2 =flowtube_defaults.dens_McorK_fact * my_square( tan( my_pi * pd / 360.0 ));  tmp =my_square(E2+(E2−D2)*t2)/(E2+(D4−E4)*t2*c2)/(D2+(D4−E4)*(1.0+t2)*c2); return ( c2 * tmp) ; } /* Protected form of sqrt for use in correcteddensity calculation Halts transmitter if attempts to do square root ofnegative number Replace before production use */ double my_squareroot(double x ) {  if ( x<0 )  {   send_message(“Fatal error in densitycorrection coefficients or equation”,1,1);   exit(1);  }  return (sqrt(x) ); } void calculate_density_basic (meas_data_type * p,meas_data_type * op, int validating) {  /* foxboro variables */  doublec2, Tz, z1, z2, z3, z4, z5, z6;  /*correction variables*/  double c1,D2, D4, E2, E4;   Tz = p->temperature_value − 20.0;   c2 = p->n_freq;   c2 = (c2 * c2) / 256.0;    p->raw_dens = (DK1 * Tz + DK2) / c2 +DK3 * Tz + DK4; /* Use density correction for flow method if enabled */   if ( do_dens_corr_flow == 1 )    {    /* temperature correct densityconstants, including correction for Tdiff */     if ( do_dens_corr_Tdiff== 1 )      Tz += flowtube_defaults.dens_Tdiff_fact *      (p->temperature_value − p->fluid_temperature ) ;     D2 = DK2 + DK1*Tz;    D4 = DK4 + DK3*Tz;     E2 = EK2 + EK1*Tz;     E4 = EK4 + EK3*Tz;    c1 = omegals( c2, D2, D4, E2, E4);     /* Calculate density withoptional offset applied by bias term*/     p->density_value =dens_flow_bias + 0.5 *      (      (D4 + E4) + my_squareroot( 4.0 * D2 *E2 / (c2*c1) + my_square(D4− E4) )      );    }    else    {    p->density_value = p->raw_dens;    if (do_dens_corr_pres == 1)    {    p->density_value = p->density_value * ( 1.0 +     flowtube_defaults.dens_pressure_fact * ( p->pressure −calibration_pressure ) );    } void calculate_massflow_basic(meas_data_type * p,       meas_data_type * op, int validating) {#define sqr(x) ((x)*(x))  double Tz;  double noneu_mass_flow;  doublepd;  /* Flow correction variables */  double km,kmd,m,d;  /* check fornon-standard conditions */  if (measurement_take != NORMAL_MEASUREMENT)  {    abnormal_massflow (p, op, validating);    return;   }  // Computemassflow  // ----------------   pd = p->phase_diff; ...    /* calculatenon-engineering units mass flow */    if (amp_sv1 < 1e−6)    noneu_mass_flow = 0.0;    else     noneu_mass_flow = 2.0 * tan (my_pi * pd / 360.0 ); // latest 2*tan (phase/2) option    // convert toengineering units    Tz = p->temperature_value − 20.0;    p->raw_mass =200.0 * flow_factor * 16.0 * (FC1 * Tz + FC3 * Tz * Tz + FC2)     *noneu_mass_flow / p->v_freq;    if (do_flow_corr_flow == 1)    {     /*Flowtube constants */     km = flowtube_defaults.flow_McorK_fact;     m= tan ( my_pi * phase_stats->getMean( ) / 360.0 );     p->massflow_value= p->raw_mass * (1.0 + km * m * m );    }    else    {    p->massflow_value = p->raw_mass;    }    if (do_flow_corr_pres == 1)   {     p->massflow_value = p->massflow_value *      ( 1.0 +flowtube_defaults.flow_pressure_fact *      ( p->pressure −calibration_pressure ) );    } /* mass flow compensation for temperaturedifference */    if (do_flow_corr_Tdiff == 1)    {     p->massflow_value= p->massflow_value *      ( 1.0 + flowtube_defaults.flow_Tdiff_fact *     ( p->temperature_value − p->fluid_temperature ) );    }

A number of implementations have been described. Nevertheless, it willbe understood that various modifications may be made. Accordingly, otherimplementations are within the scope of the following claims.

1. A method comprising: inducing motion in a conduit such that the conduit oscillates in a first mode of vibration and a second mode of vibration, the first mode of vibration having a corresponding first frequency of vibration and the second mode of vibration having a corresponding second frequency of vibration, wherein the conduit contains a fluid; determining the first frequency of vibration; determining the second frequency of vibration; and calculating a property of the fluid based on the first frequency of vibration and the second frequency of vibration.
 2. The method of claim 1, wherein the property of the fluid is the density of the fluid.
 3. The method of claim 2, wherein the density is calculated using: ${\rho_{0} = {\frac{1}{2}\left( {D_{4} + E_{4} + \sqrt{\frac{4D_{2}E_{2}}{\omega_{1}^{2}\omega_{2}^{2}} + \left( {D_{4} - E_{4}} \right)^{2}}} \right)}},$ where ω₁ is the first frequency, ω₂ is the second frequency, and D₂, D₄, E₂, and E₄ are calibration constants related to physical properties of the conduit.
 4. The method of claim 3 further comprising: determining a first temperature and a second temperature; and determining the calibration constants based on the first and second temperatures such that the calculation of the density is compensated for the affect of a differential between the first temperature and the second temperature.
 5. The method of claim 1, wherein the property of the fluid is the mass flow rate of the fluid.
 6. The method of claim 5 further comprising: measuring the oscillation of the conduit at a first location along the conduit and at a second location along the conduit; determining a phase difference based on the oscillation at the first location and the oscillation at the second location; and wherein the mass flow rate is calculated using: ${Mfact} = \frac{\left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)}{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)_{nom}}$ ${{\overset{.}{m}}_{correcrted} = {{{Mfact} \cdot {K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}_{nom} \cdot \frac{1}{\omega_{2}}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {k_{m}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}},$ where ω₁ is the first frequency, ω₂ is the second frequency, ω₁₀ is the first frequency at a zero-flow rate, ω₂₀ is the second frequency at the zero-flow rate, k and k_(m) are calibration constants related to physical properties of the conduit, and ω is the phase difference.
 7. The method of claim 1 further comprising: receiving at least one signal from a sensor configured to monitor the oscillation of the conduit; and wherein determining the first frequency and the second frequency includes determining the first frequency and the second frequency based on the at least one sensor signal.
 8. The method of claim 1 further comprising: determining physical properties associated with the conduit; determining an expression associated with one of the first frequency or the second frequency based on the physical properties; and using the expression to determine one of the first frequency or the second frequency.
 9. The method of claim 1, wherein the first frequency is determined based on the second frequency by calculating: ${\omega_{1}^{2} = {\omega_{2}^{2}\frac{\left( {E_{2} + {\left( {E_{2} - D_{2}} \right)t_{2}}} \right)^{2}}{\left( {E_{2} + {\left( {D_{4} - E_{4}} \right)t_{2}\omega_{2}^{2}}} \right)\left( {D_{2} + {\left( {D_{4} - E_{4}} \right)\left( {1 + t_{2}} \right)\omega_{2}^{2}}} \right)}}},$ where ω₁ is the first frequency, ω₂ is the second frequency, t₂=k tan² $t_{2} = {k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}$ where φ is the phase difference k, D₂, D₄, E₂, and E₄ are calibration constants related to physical properties of the conduit.
 10. The method of claim 1, wherein the first mode is a Coriolis mode, and the second mode is a driven mode.
 11. A flowmeter comprising: a vibratable flowtube, the flowtube containing a fluid; a driver connected to the flowtube and configured to impart motion to the flowtube such that the conduit oscillates in a first mode of vibration and a second mode of vibration, the first mode of vibration having a corresponding first frequency of vibration and the second mode of vibration having a corresponding second frequency of vibration; a sensor connected to the flowtube and configured to sense the motion of the flowtube and generate a sensor signal; and a controller to receive the sensor signal and configured to: determine the first frequency of vibration, determine the second frequency of vibration, and calculate a property of the fluid based on the first frequency of vibration and the second frequency of vibration.
 12. The flowmeter of claim 11, wherein the property of the fluid is the density of the fluid.
 13. The flowmeter of claim 12, wherein the density of the fluid is calculated using ${\rho_{0} = {\frac{1}{2}\left( {D_{4} + E_{4} + \sqrt{\frac{4D_{2}E_{2}}{\omega_{1}^{2}\omega_{2}^{2}} + \left( {D_{4} - E_{4}} \right)^{2}}} \right)}},$ where ω₁ is the first frequency, ω₂ is the second frequency, and D₂, D₄, E₂, and E₄ are calibration constants related to physical properties of the conduit.
 14. The flowmeter of claim 13 further comprising: a first temperature sensor configured to determine a first temperature; a second temperature sensor configured to determine a second temperature; and wherein the controller is further configured to: determine the calibration constants based on the first and second temperatures such that the calculation of the density is not affected by a temperature differential between the first and second temperatures.
 15. The flowmeter of claim 10, wherein the property of the fluid is the mass flow rate of the fluid.
 16. The flowmeter of claim 15, wherein the controller is further configured to: determine the oscillation of the conduit at a first location along the conduit and at a second location along the conduit; and determine a phase difference based on the oscillation at the first location and the oscillation at the second location, wherein the mass flow rate is calculated using: ${Mfact} = \frac{\left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)}{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)_{nom}}$ ${{\overset{.}{m}}_{corrected} = {{{Mfact} \cdot {K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}_{nom} \cdot \frac{1}{\omega_{2}}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {k_{m}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}},$ where ω₁ is the first frequency, ω₂ is the second frequency, ω₁₀ is the first frequency at a zero-flow rate, ω₂₀ is the second frequency at the zero-flow rate, k and k_(m) are calibration constants related to physical properties of the conduit, and φ is the phase difference.
 17. The flowmeter of claim 10, wherein the controller is further configured to determine the first frequency and the second frequency based on the sensor signal.
 18. The flowmeter of claim 10, wherein the controller is further configured to; determine physical properties associated with the conduit; determine an expression associated with one of the first frequency or the second frequency based on the physical properties; and use the expression to determine one of the first frequency or the second frequency.
 19. A flowmeter transmitter comprising: at least one processing device; and a storage device, the storage device storing instructions for causing the at least one processing device to: induce motion in a conduit such that the conduit oscillates in a first mode of vibration and a second mode of vibration, the first mode of vibration having a corresponding first frequency of vibration and the second mode of vibration having a corresponding second frequency of vibration, wherein the conduit contains a fluid; determine the first frequency of vibration; determine the second frequency of vibration; and calculate a property of the fluid based on the first frequency of vibration and the second frequency of vibration.
 20. The transmitter of claim 19, wherein the property of the fluid is the density of the fluid.
 21. The transmitter of claim 20, wherein the density of the fluid is calculated based on: ${\rho_{0} = {\frac{1}{2}\left( {D_{4} + E_{4} + \sqrt{\frac{4D_{2}E_{2}}{\omega_{1}^{2}\omega_{2}^{2}} + \left( {D_{4} - E_{4}} \right)^{2}}} \right)}},$ where ω₁ is the first frequency, ω₂ is the second frequency, and D₂, D₄, E₂, and E₄ are calibration constants related to physical properties of the conduit.
 22. The transmitter of claim 21 further storing instructions for causing the at least one processing device to: determine a first temperature and a second temperature; and determine the calibration constants based on the first and second temperatures such that the calculation of the density is not affected by a temperature differential between the first and second temperatures.
 23. The transmitter of claim 19, wherein the property of the fluid is the mass flow rate of the fluid.
 24. The transmitter of claim 23 further storing instructions for causing the at least one processing device to: determine the oscillation of the conduit at a first location along the conduit and at a second location along the conduit; and determine a phase difference based on the oscillation at the first location and the oscillation at the second location, wherein the mass flow rate is calculated based on: ${Mfact} = \frac{\left( {{\frac{\omega_{2}^{2}}{\omega_{1}^{2}}\left( {1 - {2k\quad{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)} - 1} \right)}{\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)_{nom}}$ ${{\overset{.}{m}}_{corrected} = {{{Mfact} \cdot {K\left( {\frac{\omega_{20}^{2}}{\omega_{10}^{2}} - 1} \right)}_{nom} \cdot \frac{1}{\omega_{2}}}{{\tan\left( \frac{\varphi}{2} \right)} \cdot \left( {1 + {k_{m}{\tan^{2}\left( \frac{\varphi}{2} \right)}}} \right)}}},$ where ω₁ is the first frequency, ω₂ is the second frequency, ω₁₀ is the first frequency at a zero-flow rate, ω₂₀ is the second frequency at the zero-flow rate, k and k_(m) are calibration constants related to physical properties of the conduit, and φ is the phase difference.
 25. The transmitter of claim 19 further storing instructions for causing the at least one processing device to determine the first frequency and the second frequency based on the sensor signal.
 26. The transmitter of claim 19 further storing instructions for causing the at least one processing device to: determine physical properties associated with the conduit; determine an expression associated with one of the first frequency or the second frequency based on the physical properties; and use the expression to determine one of the first frequency or the second frequency. 